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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Integration and Differentiation In Exercises 1- 6, verify the statement by showing that the derivative of the right side is equal to the integrand on the left side.

( 4 x 3 1 x 2 ) d x = x 4 + 1 x + C

To determine

To prove: The verification of the indefinite integral (4x31x2)dx=x4+1x+C such that the derivatives of the right side is equal to the integrand on the left side.

Explanation

Given Information:

The provided indefinite integral is (4x31x2)dx=x4+1x+C.

Formula used:

Power rule of derivative is ddx(xn)=nxn1, where n is a real number.

Sum rule for function f(x)=u(x)+v(x), where f(x) and g(x) are differentiable functions of x, then ddx[f(x)+g(x)]=f(x)+g(x).

Proof:

Consider the indefinite integral (4x31x2)dx=x4+1x+C.

The left side integrand of the indefinite integral expression is (4x31x2).

The right-side of the provided indefinite integral expression is (x4+1x+C).

Differentiate the right-side term with respect to x, use the sum rule of differentiation ddx[f(x)+g(x)]=f(x)+g(x)

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